Advanced Calculus Single Variable

11.4 Dini’s Criterion For Convergence

Fourier series like to converge to the midpoint of the jump of a function under certain
conditions. The condition given for convergence in the following theorem is due to Dini. It is a
generalization of the usual theorem presented in elementary books on Fourier series methods.
[3].

As pointed out by Apostol [3], where you can read more of this sort of thing, this is a very
remarkable result because even though the Fourier coeficients depend on the values of the
function on all of

[− π,π ]

, the convergence properties depend in this theorem on very local
behavior of the function.